Macromolecules 1993,26, 3282-3288
3282
The Effect of Swelling on the Elasticity of Rubber: Localization Model Description Jack F. Douglas’ and Gregory B. McKenna Polymers Division, National Institute of Standards & Technology, Gaithersburg, Maryland 20899 Received August 28, 1992; Revised Manuscript Received March 9, 1993
ABSTRACT: The success of the localization model in describing the elasticity of unswollen natural rubber is reviewed and the model is extended to describe the elasticity of swollen networks. In contrast to the Frenkel-Flory-Rehner modeling of network elasticity, the localization model predicts that the mechanical response of swollenrubbers changesqualitatively in going from low to high cross-linkdensities. The implications of these predictions are discussed. 1. Introduction
2. Review of the Localization Model
Frenkell and Flory and Rehner2 introduced some fundamental hypotheses relating to the swellingof rubber and the changes of rubber elasticity associated with this swelling,viz., (1)the free energy of mixing and the elastic free energy in swollen networks are additive and ( 2 ) the elastic strain energy density function is invariant to swelling. McKenna et a L 3 s 4 recently evaluated these hypotheses for samples of dicumyl peroxide cross-linked natural rubber. These experimentalstudies indicated that if the dry rubber free energy density was determined from continuum mechanics theory, rather than from a “molecular” model, then the swollen state compressive p r o p erties could be predicted directly from the FFR hypotheses. Recently, we have realized that it is difficult to similarly explain the change of rubber elasticity with swellingindicated in the experimentsof Gumbrell, Mullins, and Rivlin5and Allen et al.6 Further examination of the FFR hypotheses seems warranted, especially the second hypothesis. In previous work7on characterizing the elasticity of dry rubber we found that the localization model described the dry rubber data well and yielded reasonable magnitudes of the model parameters. This previous work will be summarized as a preliminary to an extension of the model to describe network swelling. The extension of the localization model offers a resolution of the apparent ~ inconsistency between the McKenna et a l . 3 1compression results and the extension results of Gumbrell, Mullins, and Rivlin5and Allen et a1.6 in that it predicts two regimes of behavior. At high cross-linking densities the elastic strain energy function is invariant to swelling and, thus, accords with the second Frenkell-Flory-Rehner2 (FFR) hypothesis. However, a t low cross-link densities the model predicts that the second FFR1>2hypothesis no longer holds. In the following section we summarize prior results showingthe success of the localization model in describing the dry state properties of peroxide cross-linked natural rubber. We then present the discrepancy between the swollen state compression results of McKenna et a1.394~~ and the extension results of Rivlin et al.6 and Allen et al.6 In the penultimate section we present an extension of the localization model to describe swollen rubber and we discuss the implications of the model in comparison with experiment. The apparent breakdown of the second FFR hypothesis under certain conditions is perhaps the most important implication of the localization model.
Molecular models of polymeric network elasticity have concentrated on certain minimal aspects of rubber networks. The classical theories focused on the property of network connectivity. Recent theories have stressed “nonideal” contributions to network elasticity. For example, Edwards and co-worker~~*~ have emphasized “topological”interactions associated with the uncrossability of network chains. DiMarziolO and Jackson et al.11 attempted to derive more realistic models by including the chain packing effects in a mean-field approximation. Gaylord and D o ~ g l a s incorporated ~~J~ these minimal features of a cross-linked network (network connectivity, entanglement interaction, finite chain volume) into a simple theory of rubber elasticity which they call the “localization model”. In this model the change in the network free energy with deformation is given by12J3 3
AF(elastic)/V, = (G,/2) le1
3
- 1)+ G , C ( h i - 1)
(1)
Fl
where AF(e1astic) is the elastic contribution to the free energy of the network, VOis the volume of the dry rubber sample, and the hi are the principal extension ratios (“stretches”). The variables G, and G, are defined below. The first term in eq 1 is the classical network chain connectivity contribution and the second term is the “localization” contribution. Derivation of eq 1 assumes network in~ompressibility.~2J3 In the absence of localization interactions (G, = 0), eq 1reduces to classical rubber elasticity theory. Although various predictions have been made for the magnitude of the network connectivity modulus G,, it is proportional to the cross-link density i G , = C,;kbT (2) in all of the classical models. Co is a constant related to network structure, k b is the Boltzmann constant, and T is the absolute temperature. For an end-linked tetrafunctional network there are various estimates of Real polymer networks contain dangling ends and various inhomogeneities and the contributions of these imperfections to COare described by Flory.15 (Dossin and Graessley’g argue that discrete local “entanglements” may contribute to COso that even in principle an exact specification of the chemical crosslinking might be insufficient to determine COand they therefore take Co to be a variable determined by exper-
This article not subject to U S . Copyright. Published 1993 by the American Chemical Society
Macromolecules, Vol. 26, No. 13, 1993 iment.) Limited specification of network structure also causes some uncertainty in CO. In the experimental comparisons below we will take COequal to the James and Guth estimate17CO= 1/2,lackingperfect informationabout the network structure. Recent experiments by Gnanou et al.18 suggest this to be a reasonable presumption. The localization parameter Ge of eq 1accounts for the restriction of the configurations of a network chain by surrounding network chains. The basic effect of having a chain “hemmed in” by its surroundings is to reduce the average number of chain degrees of freedom. The physical picture of the network chain ‘localized” to an irregularity shaped region, defined by the interaction of the network chain with surrounding chains, is similar to Edwards’ original formulationof a tube model of network e1asti~ity.l~ Gaylord and Doug1asl2J3take this type of model further by arguing that the “tube” radius reflects the hard-core cross-sectional radius of the polymer and thus the tube volume in which the chain is localized is on the order of the chain molecular volume. Since the chain molecular volume is inuariant to a macroscopic deformation, they argue that the tube volume should also be an invariant. From this constraint they estimate the variation of the chain localization length as a function of deformation to obtain eq 1. Gaylord and Douglas13find that the localization modulus Ge has two contributions
G, = 6(;kT) + GN*
(3) where GN*is the cross (cross-link independent) plateau modulus of the polymer melt and 6 is a constant. Another implication of the localization model is a rough inverse relation between the melt plateau modulus GN*and the average chain cross sectional area u. This inverse relation also implies that the melt entanglement molecular weight M eis proportional to the average chain cross sectional area. There are some experimental data which support these relations,20 i.e., GN* = l / a and Me = cr. Further measurements are necessary to test their quantitative validity. In the next section we describe our comparison of torsional data for cross-linked natural rubber with the localization model predictions. Some details of the methods of analysis in this comparison are given in Appendix A. 3. Comparison of the Localization Model with Dry Networks Torsional Data The localization model was fitted successfullyto torsion data and to uniaxial and biaxial extension data7J2on dry rubber letting both G, and Ge be free parameters. In each case the quality of fit was good and the magnitudes of the parameters G, and G e were found to be reasonable. However, a real test of the theory requires a determination of whether the molecular parameters G, and Ge vary as predicted. Specifically, we can set G, to the value estimated from the James and Guth network theory17and observe how well we can fit the elasticity data with a single adjustable parameter Ge. The localization model further constrains the variation of Ge so that it is not a “free” parameter. Equation 3 implies that G e reduces to the plateau modulus of the un-cross-linked rubber in the formal limit v 0 and that Ge should increase linearly with cross-link density. (The limit 5 t0 is formal since the equilibrium modulus will vanish for Y sufficientlysmall that the rubber is a liquid.) The model parameters of the theory are evidently highly constrained in terms of direct observables. The only unspecified parameter in the theory
-
Elasticity of Rubber 3283 Table I. Values of G. Obtained for Localization Model Determined from Curve Fitting of Equation A.4 to Natural Rubber Torsional Data c n = 112 rubber sample (moUcms) G, (Pa) G. (Pa) APHRl 2.09 X 106 2.58 X 10.‘ 5.94 X 106 APHR2 APHR3 APHR5 APHR7.5 APHRlO APHR15
5.22 X 8.35 X 1.46 X 2.24 X 3.03 X 4.59 X
106 106 lo-‘ lo-‘ lo-‘ lo-‘
6.44 X 1.03 X 1.80 X 2.76 X 3.74 X 5.66 X
10.‘ lo6 106 106 106 106
7.24 X 1.09 X 1.67 X 2.71 X 2.33 X 3.58 X
106 10s 10s
10s 10s 10s
is the constant of proportionality in the dependence of Ge on .; McKenna et ala7have recently compared the predictions of the localization model with experimental torsion data on dicumyl peroxide cross-linked rubber in the dry state. In the following paragraphs we briefly review this comparison. Cross-link densities of their samples are reported in Table I based on the assumption that each dicumyl peroxide molecule decomposes to form a tetrafunctional cross-link. McKenna et al.7 performed least-squares fits of the derivatives of eq 1 to the torsional data for each cross-linkdensity assuming that CO= 1/2. (In the original references the reader will see that the fits are very good.) The corresponding estimates of G, are included in Table I based on this James and Guth17 estimate for CO. The curve fits then provide the means to calculate Ge from eq 2 and in Table I it can be seen to increase with cross-link density and that the localization contribution (Ge) to the rubber elasticitypredominates over the classical term (G,). Dossin and GraessleylG and Oppermann and Rennar21 make similar observations regarding the elasticity of polybutadiene and poly(dimethylsi1oane) networks, respectively. A linear regression on the data in Table I shows that Ge is well represented by a linear function of crosslink density Ge(5)= 5.56 X lo5 (Pa) + 6.83 X
lo9 (Pa cm3/mol)
(4) as predicted by the localizationmodel. Moreover,the value of Ge(i) extrapolated to zero cross-link density is close to the reported value of the melt plateau modulus, viz, G*N = 5.77X lo5(Pa). This also agrees with localization model predictions. The agreement between the model and experiment exhibited by these measurements is encouraging. 4. Compression and Extension of Swollen Networks-An Apparent Anomaly In recent work, McKenna et al.a4 were able to use the dry rubber strain energy density function obtained from torsion measurements to calculate the compression response of the rubber in the swollen state by applying the concepts of the Frenkel-Flory-Rehner2 hypothesis and the Valanis-Lande122 continuum equations of rubber elasticity (see Appendix A). In terms of the volume fraction I$ of rubber and the elastic contribution to the Helmholtz free energy AF,they were able to write
- “22 CY2 - l l a “11
“R=-=
Macromolecules, Vol. 26, No. 13, 1993
3284 Douglas and McKenna
2.00
t h
ad
Y
3.0
1
1.75
i L
1.50 v)
s
i
i
M
u”
1.25 1 .oo
0.50 0.75
1 .oo
0.00 0.25 1 .o 0.40
0.50
0.60
0.70
0.80
0.90
1.00
l/a
(true) stress difference in the swollen rubber, and a is the mechanical deformation superimposed on the swollen rubber. P ( x ) = dAF/dx and & = 4-1/3 is the swelling deformation. Equation 6 correctly predicted the swollen state response of the cross-linkednatural rubbers described in Table I and for solvents having a range of swellingpower from poor (acetone) to good (benzene), although some deviation in behavior was apparent at extremely high crosslink densities. Because eq 6 successfully describes the swollen state compressive properties of natural rubber networks3s4and because eq 1 had been found to describe the dry state properties of the rubber, the implication is that the combination of eq 1and eq 6 would successfully describe the behavior of swollen networks in simple extension. McKenna et aL3v4did not perform experiments in simple extension and, therefore, we were unable to make a direct comparison of results and theory. However, for the present illustrative discussion we can use the experimentally determined free energy function of McKenna et a l S 3 p 4 (dry state data) to calculate a theoretical stress-strain behavior for swollen natural rubber in simple extension based on the FFR hypotheses (eq 6). The result of such a procedure is shown in Figure 1 in which reduced stress is plotted against l / a over a range of rubber volume fractions 4. The values of C$ are similar to those considered by Gumbrell et al.6 and Allen et al.6 in their extension experiments on cross-linked and swollen natural rubber. We see in Figure 1 that the slope of the reduced stress predicted from eq 6 (FFR hypothesis, dry state determined strain energy function) decreases with increasing degree of swelling, as generally observed. However, the decrease is significantly less than that observed in the studies of refs 5 and 6 (see also Treloar24 for a summary of this classic set of data). It is conventional to examine deviations from classical rubber elasticity theory in uniaxial extension through the introduction of Mooney-Rivlina parameters. For a swollen rubber we write the reduced stress = I(a,4) = r0/[4-”3(a- l/aZ)l
0.25
0.50
0.75
1 4
Figure 1. Modified reduced stress 4-1/3a~vs l / a for swollen natural rubber calculated from the experimentally determined strain energy function for sample APHR1, in conjunction with the FrenkelLFlory-Rehner2hypotheses. Lines represent leastsquares fits to Mooney-Rivlin function. (Seetext for discussion.)
CR4-1’3
0.00
(7)
I(cr,4)= C,(4) + C&$)/a (8) where 70is the stress relative to the dry, undistorted rubber, 4 is the volume fraction of the rubber, and a is the
Figure 2. Mooney-RivlirP constant Cz from modified reduced stress vs 1- 4 for swollennatural rubber. Pointsrepresent results calculated from Figure 1 Mooney-Rivlin fits. Curve through points represent Cz 0: 41/3. Straight line is representative of literature data for which CZa 4l (or4‘4. (Seetext for discussion. Reader is also referred to Allen et al.,B Price,n and TreloaP for extended discussions of the value of the exponent.)
mechanical deformation in the swollen state. Equation 8 is empirical and applies only to extension measurements. Nonetheless, it provides a useful means to characterize the elasticity of rubbers if employed properly. We note here that C1 and Cp are functions of volume fraction. When Cp is obtained by fitting straight lines to the predicted results shown in Figure 1the result is that CZvaries as (b1/3 as shown in Figure 2. This is dramatically different from the exponentM(Cz 4’ or 44/3)often observedin extension experiments on swollen rubber (see section 5). This weaker 4 dependence of Cp from previous6p6experimental studies is depicted in Figure 2. The large discrepancy for the concentration dependence of Cp calculated from eq 6 and the free energy function data of McKenna et a L 3 t 4 (which describes the compressive behavior in the swollen state) and the observations of Gumbrell et alS6and Allen et al.6 is difficult to rationalize in terms of experimental errors. A possible explanation is that one or both of the FrenkelFlory-Rehner2 hypotheses are incorrect. Since the compression results of McKenna et ala4 are consistent with the FFR hypotheses, outright rejection of the FFR hypotheses is difficult. On the other hand, the data of Gumbrell et aL5and Allen et aL6were obtained in uniaxial extension and seem inconsistent with the FFR modeling. Could there be something fundamentallydifferent between compression and extension? We see no reason to believe this, but recognize this as a possibility. Another possibility is that there is a difference between the samples used in ~ those used by the experiments of McKenna et a l . 3 ~and Gumbrell et aL5and Allen et al.6 In the case of the former the samples were highly cross-linked while in the latter two studim the degree of cross-linkingwas lower (seeTable 11). There is some overlap in the ranges, however. In the following section the localization model is extended to swollen rubber and a possible explanation for the discrepancy between the sets of data is discussed.
-
5. Extension of the Localization Model to Swollen Network Elasticity The swellingof dry polymer networks is a rather unique phenomenon. Lightly cross-linked dry networks can absorb large quantities of certain fluids and increase their volume by orders of magnitude without appreciable change
Macromolecules, Vol. 26, No. 13, 1993
Elasticity of Rubber 3285
Table 11. Comparison of Dmea of Swelling of Natural Rubber in Benzene for Samples Used in Studies of Gumbrell et al.6 and McKenna et al.84 study Gumbrellet al.5
sample designation A
B C D E F G
McKenna et al.3.4
APHRl APHR2 APHR3 APHR5 APHR7.5 APHRlO APHRl5
4
116 5.21 5.52 4.95 4.26 4.48 3.89 3.68 6.32 4.78 3.91 2.99 2.36 2.47 1.94
0.192 0.181 0.202 0.235 0.223 0.257 0.272 0.158 0.209 0.256 0.334 0.423 0.405 0.515
of shape. The cross-links give these highly swollen materials a remarkable rigidity. Theoretical prediction of the elasticity of a swollen network from a knowledge of the dry network elasticity is a challenging, unsolved problem. Flory acknowledgedz5 that the predictions of the classicalrubber elasticity theory are disturbingly inaccurate and suggested that the cause for the observed deviations is also responsiblefor deviations from classical rubber elasticity theory in the dry state. Consider a sample of dry cross-linked rubber of volume Vowhich is subjected to an isotropic swelling of dimensions by a factor A,. The swollen rubber of volume V is then deformed uniaxially and we measure the stress as a function of swelling and the mechanical deformation ai. Extension of eq 1 to describe the elasticity of a swollen network is by no means obvious, but we can progress by following arguments given previously by FFR. The mechanical stretch parameter aj in the FFR theory is replaced by A, ai under the presumption that a “stretch is a stretch” regardless of the mode of extension (swelling or mechanical). In the spirit of the FFR theory we replace Xi in eq 1 by ai& to obtain a
3
f(X,)
(9)
The unspecified function f ( X , ) occurs because swelling is a nonconstant volume deformation (see Appendix B). Assuming network incompressibility allows us to neglect f(&)in the consideration of the swollen network elasticity where only the derivative of AF(ai,&) with respect to ai is required. The free energy density of the swollen network is obtained by multiplying (9) by the polymer volume fraction 9 as
To estimate the concentration dependence of G, we recall eq 3, = G N ( 4 = 1) + 6(4 = 1);kBT (11) The variation of the plateau modulus GN with concentration is estimated using the localization model in the absence of cross-links. Packing arguments similar to those used to estimate the nonclassical term in eq 3 give12J3 G,&1
-
GN(4) GN*rb2, 4 = 1 (12) The derivation of eq 12 includes the factor of 4 indicated in eq 10 and a generalization of eq 12 to d-dimensions is given in Appendix A. This generalization indicates that the origin of the concentration dependence in eq 12 does not arise from binary contacts. Rather it arises from chain packing. Equation 12 accords well with experimental observations on concentrated high molecular weight polymer solutions.26 Our generalization of (11) involves replacing G N ( =~ 1) by (12) to obtain (see Appendix A, eq A.5d) G,$ Z [GN*~]I$+ 6(4 1) (ikBZ‘)4, 4 E (13) The factor of 4 in the second term comes simply from the change in rubber volume with swelling as in the classical term G, in (10). Note that the 6 term in (11) is simply proportional to G, so it seems natural to treat the 6 term in (13) in a similar fashion. Once G& is fixed as in (13) there are no free parameters in our predictions of the elastic deformation of swollen networks based on (10). We now compare the results of this generalized elasticity model with the swellingexperiments mentioned above for lightly cross-linkedand highly cross-linkedrubbers. Observe that the plateau modulus term and cross-link-dependent term in (13) have different concentration dependencies. This difference could have important implications for the elastic properties of swollen rubbers. For a uniaxial stretch we define the deformation parameters (A1 = a,XZ = d 2 , A3 = a-1/2) and calculate the tension per unit area as26 7 = d[AF/VJ/da (14) Inserting (10) with G, defined by (31, (ll),and (13) into (14) gives the tension
T
= G,41/3(~ - I/&
(15)
A generalization of this expression to d-dimensions is given in Appendix A. The tension 70 relative to the dry state cross-sectional area is related to rF5 70 = 7/42/3,and the tension is sometimes written in the “reduced form” shown in eq 7. A deviation of I(a,4) from constancy at a fixed temperature is a measure of the “nonideal” contribution to the rubber ela~ticity.~The reduced tension in (7) is investigated in detail by Gumbrell et al.5 for uniaxial extension of swollen rubber and (7) is introduced to compare with these data. Inserting (15) into the definition of gives G ~ * f $ ] f $ ~/ ~c(~l ~ ’ ~ ) / aa“()l (16) For a large uniaxial extension a >> 1 we have (1 - ( ~ - ~ / ~ ) / ( l - a-9 = 1 so that I(a,&) reduces to
I((Y,+,) = G,
The James and Guth” network theory should afford a reasonable estimate of G,, as discussed in section 2, and this value can be compared with estimates of G, determined from elasticity measurements. G, should be independent of 9 and proportional to the network cross-linking density in comparisons of (10) to experiment. The specification of the concentration dependence of G, is a subtle question on which our extension of the localizatioin model pivots.
[6(;k~T) + G ~ * 4 ] @ ~ ’-~ ( 1
[6;k,T
I(a,4)
- G, +
&$)/a,
a >> 1
(17)
B(4) = ( 6 ; k ~ T+ GN*I$)$~’/~ (18) This is only a formal limit since finite extensibility effects
3286 Douglas and McKenna
Macromolecules, Vol.26,No.13,1993
become significant for large a. Equations 17 and 18 are introduced for qualitative comparison with experimental studies in the discussion below. Gumbrell et al.5, Allen et aL6, and Price27 find an empirical correlation of I(&) identical to (17) for uniaxial extension of swollen rubbers. Equation (17) is the Mooney-Rivlin equation (see eq 8 and Appendix A) and this result is actually derived from the localization model along with a specification of the concentration dependence of C1 and CZ.Equation 8applies to correlations of uniaxial extension data in a limited range of deformations. Again we emphasize that the Mooney-Rivlin correlation does not apply to compression and other geometries of deformation and the localization model does not reduce to this simple form in other types of deformation. Experimental values of C1($)obtained by Gumbrell et al.5 varied very slowly with polymer fraction 4 and experiments by Allen et al.6indicate that Cl(4)is“virtual1y independent of the degree of dilution”. Similarly, examination of eq 16 shows that in the limit of infinite extension I(a,&)reduces to G, independent of the degree of swelling. G, is a network invariant which apparently can be measured with fair precision. In contrast to C1,the CZparameter in (8) is found to have a rather strong concentration dependence in the rubbers employed by Gumbrell et al.,5 Allen et al.,6 and Prices2’ Gumbrell et a l . 5 f o ~ dthat C2(9)for many rubbers (varying cross-link conditions) varied in direct proportion to polymer volume fraction, C2 a 4, to a good approximation. This variation was found to be independent of solvent. Later, experiments by Mullinem and Pricez7gave the empirical correlation, C2 a 4413, as a slight improvement over the initial correlation of Gumbrell et aL5 Before comparing with the predictions of the localization model, we emphasize that the large changes of the polymer volume fraction at swelling equilibrium necessitate rather light cross-linking in these rubbers. It is possible that different behavior would be observed in highly cross-linkednetworks where only modest swelling is possible. For relatively low cross-linking the &dependent term in (17) and (19) is neglected so that the concentrationdependent factor B ( d ) , corresponding to Cz(4) in (8), reduces to
G,
B(4)= GN* gj4/‘ (low cross-link density, uniaxial extension) (19) Equation 18 accords well with the experimental observations of Mullinsm and Price27 for the variation of Cz(4) in extension. However, the cross-link-dependent contribution to B(4) in (17) should dominate for high crosslinking densities to give B(+) = b(;kB7)$1/3 (high cross-link density, extension) (20)
This is qualitatively different behavior from eq 19 (see Figures 4 and 5 ) . If we trace this result back to the original form of the free-energy density function AF(ai,&),we find that for high cross-link densities AF is given by eq 10 with the high cross-link density limiting form of eq 11
i=l
-
6&,T
0:
G,
(22)
The form of the free energy density in (21) is nonideal and invariant to swelling up to the 4 scale factor and the f(&) term. As described in the previous section, such behavior is calculated from the parameters obtained for cross-linked natural rubber using the Valanis-LandeP function and calculatingthe tensile behavior of swollen rubber using eq 6.
This different concentration dependence of the nonideal contribution to the rubber elasticity in high cross-link and low cross-link density rubbers is a striking prediction of the localization model. The variation of the AF with concentration seems to depend on the extent of network cross-linking. Further experimental studies are needed to check this interesting prediction. The discussion above assumes that the networks are cross-linked in the melt. The localization model suggests that networks of different elasticity might be obtained for networks synthesized from solution. Erman and Mark29 discuss the effect of varying the concentration at the time of cross-linking on G,. These arguments indicate that CO in eq 2 picks up a dependence on the volume fraction at which the cross-linking occurs. The localization parameter G, is also expected to reflect the network density at the state of cross-linking. It seems reasonable to replace the “entanglement” GN contribution to G, by under the assumption that network drying is not equivalent to a deformation. The & term reflects the number of network chains per unit volume under the conditions of cross-linking. Equation 23 would imply that lightly crosslinked rubbers in highly diluted solutions, & 4*42*43 We note that the ideal network theory of James and Guthl’ and the more recent Flory-Erman theory44 do not satisfy the stability condition (B.l) and this has significant consequences for the predictions of network swelling in these models.40
References and Notes (1) Frenkel, J. Rubber Chem. Technol. 1940,13,264. (2) Flory, P. J.; Rehner, J., Jr. J. Chem. Phys. 1943, 11, 521. (3) McKenna, G. B.;Flynn, K. M.; Chen, Y.-H. Macromolecules 1989,22,4507.
(Scaling of plateau modulus with concentration in variable dimension is given in Appendix B of this work.) (14) The estimation of COhas a long and colorful history which is concisely summarized by K. Dusek and W. Prins (Adu.Polym. Sci. 1969, 6, 1). See also: (a) Flory, P. J. Polym. J. 1985,17, 1. (b) Treloar, L. R. G. The Physics of Rubber Elasticity, 3rd ed.; Clarendon Press: Oxford, 1975. (15) Flory, P. J. Proc. R. SOC.London, A 1976, 351, 351. (16) (a) Dossin, L. M.; Graessley, W. W. Macromolecules 1979,12, 123. (17) James, H. M.; Guth, E. J. Chem. Phys. 1943,11,455. (18) Gnanou, Y.; Hild, G.; Rempp, P. Macromolecules 1987,20,1662. (19) Edwards, S. F. Proc. Phys. SOC.1967,92,9. We emphasize that
the original tube model of Edwards is unrelated to the reptation tube model of polymer melt dynamics. (20) (a) He, T.; Porter, R. S. Macromol. Chem., Theory Simul. 1992, 1,119. (b) Boyer, R. F.; Miller, R. L. Polymer 1976,17,1112. (c)Zang, Y.; Carreau, P. J. J.Appl. Polym. Sci. 1991,42,1965. (21) Oppermann, W.; Rennar, N. Prog. Colloid Polym. Sci. 1987,75, 49. (22) Valanis, K. C.; Landel, R. F. J. Appl. Phys. 1967,38,2997. (23) (a) Mooney, M. J.Appl. Phys. 1940,11,582. (b) Rivlin, R. S. Philos. Trans. R. SOC.London, A 1948,241, 379. (24) Treloar, L. R. G. The Physics of Rubber Elasticity, 3rd ed.; Clarenden Press: Oxford, 1975. (25) Flory, P. J. Principles ofPolymer Chemistry;Comell University Press: Ithaca, NY, 1953. (26) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; Wiley: New York, 1981. (27) Price, C. Proc. R. SOC.London, A 1976,351, 331. (28) Mullins, L. J. Appl. Polym. Sci. 1959,2, 257. (29) Erman, B.; Mark, J. E. Macromolecules 1987,20,2892. (30) Price, C.; Allen, G.; DeCandia, F.; Kirkham, M. J.;Subramaniam, A. Trans. Faraday SOC.1970,11,486. (31) (a) Johnson, R. M.; Mark, J. E. Macromolecules 1972,5,41. (b) Mark, J. E. Rubber Chem. Technol. 1975,48,495. (32) Gaylord, R. J.; Twardowski, T. E.; Douglas, J. Polym. Bull. 1988, 20, 305. (33) Kearsley, E. A.; Zapas, L. J. J. Rheol. 1980,24, 483. (34) (a) Jones, D. F.; Treloar, L. R. G. J. Phys. D 1975,8,1285. (b) Vangerko, H.; Treloar, L. R. G. J. Phys. D 1978,11,1969. (c) Kawabata, S.; Kawai, H. Fortschr. Hochpo1ym.-Forsch. 1977, 24,89. (d) Obata, Y.; Kawabata, S.; Kawai, H. J. Polym. Sci. 1970, 8, 903. (35) Penn, R. W.; Kearsley, E. A. Trans. SOC.Rheol. 1976,20, 227. (36) Rehage, H.; Veyssie, M. Angew. Chem., Engl. Ed. Engl. 1990, 29, 439. (37) Pelzer, H. Monatsh. Chem. 1938, 71, 444. (38) Wall, F. T. J. Chem. Phys. 1942, 10, 132, 485; 1943, 11, 527. (39) (a) Kuhn, W. KolZoid-Z.l936,76,258. (b) Kuhn, W. Kolloid-2. 1939,87,3. (c) Kuhn, W.; Grin, F. J. Polym. Sci. 1946,1, 183. (40) McKenna, G. B.; Hinkley, J. A. Polymer 1986, 27, 1368. (41) Freed, K. F. J. Chem. Phys. 1971,55, 5588. (42) Gee, G.; Herbert, J. B. M.; Roberta, R. C. Polymer 1966,6,541. (43) Crissman, J. M.; McKenna, G. B. Polym. Mater. Sci. Eng., in
press.
(44) (a) Florv. P. J.: Erman. B. Macromolecules 1982.15.800. (bl , F~OW, P. J. Poiym. J. i985,17,1. (45) (a) Dubault, A.; Deloche, B.;Herz. J. Macromolecules 1987.20. 2096. (b) Dubault, A.; Deloche, B.; Herz, J. Macromolecules 1990,23, 2823. -
